Chapter 8.7, Problem 2BGP

To determine

To factor: The given polynomial.

Answer to Problem 2BGP

The polynomial $2a^2+10a+25$ is a prime polynomial.

Explanation of Solution

Given information:

The given polynomial is $2a^2+10a+25$ .

Concept used:

A polynomial of the form $a{x}^2+bx+c$ can be factored by British method and a polynomial that can be written as the difference of two squares can be factored as $a^2-b^2=\left(a-b\right)\left(a+b\right)$ .

British method: For polynomial $a{x}^2+bx+c$ , find two factors of $a\times c$ such that their sum is b.

Calculations:

The given polynomial E (say) is $E=2a^2+10a+25$ .

Now, we try to find out two factors of $2\times 25=50$ such that their sum is 10 as:

  $2\times 25=50=2\times 5\times 5=10\times 5=2\times 25$ , in no case sum of factors of 50 is 10.

Also, the polynomial $2a^2+10a+25$ cannot be written as difference of two squares. So it cannot be factorized by this method.

Thus, the polynomial $2a^2+10a+25$ cannot be factorized, hence it is a prime polynomial.

Conclusion:

The polynomial $2a^2+10a+25$ is a prime polynomial.